Corollary 16.5.2861DB6FF8D92F

Proof. Step 1. We first consider the case $\psi =\phi $.

We can find $C>0$ so that $\varphi \leq \phi +C$. Therefore, $\phi -\varphi \geq -C$ almost everywhere with respect to $\theta _{\gamma _1}\wedge \cdots \wedge \theta _{\gamma _ n}$. So the integral (16.5.29) is well-defined and takes value in $\mathbb {R}\cup \{ \infty \} $. We show that the integral is finite. For this purpose, we may replace $\varphi $ by $\varphi -C$ and assume that $\varphi \leq \phi $. Put

Then by the multi-linearity of the non-pluripolar product proposition 13.11.4 6 and positivity, we have

Thanks to corollary 16.5.25, $\eta \in \mathcal{E}^1(X,\theta ;\phi )$. Choose $A>0$ so that $\eta \leq \phi +A$. Applying 16.5.26 to $\varphi ,\eta $ and $\phi +A$, we get

Hence $\int _ X(\phi -\varphi )\, \theta _{\eta }^ n<\infty $, and Step 1 follows.

Step 2. We handle the general case. Take $C>0$ so that

Then

Step 1 applied to $\varphi -C$ and to $\psi -C$ shows that $\phi -(\varphi -C)$ and $\phi -(\psi -C)$ are integrable with respect to $\theta _{\gamma _1}\wedge \cdots \wedge \theta _{\gamma _ n}$. Hence the positive and negative parts of $\psi -\varphi $ are both integrable, so (16.5.29) is well-defined and finite. $\square$